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Journal of Physics A Mathematical and General

Comment on 'Solutions of the Yang-Baxter equation for isotropic quantum spin chains'

Murray T. Batchelor, C M Yung

Published Jul 21, 1994• 5 citations
Publisher

Abstract

We comment on a recent paper by Kennedy (J. Phys. A: Math. Gen. 25 (1992) 2809) in which a systematic search for integrable spin-S su(2)-invariant quantum chains for S<or=6 revealed four spin-S families of integrable chains along with an additional integrable chain at S=3. We identify these su(2)-invariant chains with known G-invariant R-matrices, where G is a simple Lie algebra, and give arguments that Kennedy's results may well constitute the complete classification of integrable spin-S su(2)-invariant chains.

DOI
10.1088/0305-4470/27/14/028

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We comment on a recent paper by Kennedy (J. Phys. A: Math. Gen. 25 (1992) 2809) in which a systematic search for integrable spin-S su(2)-invariant quantum chains for S<or=6 revealed four spin-S families of integrable chains along with an additional integrable chain at S=3. We identify these su(2)-invariant chains with known G-invariant R-matrices, where G is a simple Lie algebra, and give arguments that Kennedy's results may well constitute the complete classification of integrable spin-S su(2)-invariant chains.
The author considers solutions of the Yang-Baxter equation such that the logarithmic derivative of the transfer matrix yields a quantum spin Hamiltonian which is isotropic in spin space, i.e. SU(2)-invariant. Four such solutions are known for each value of the spin S. (For S=1/2 they degenerate into the same solution, and for S=1 they only give three different solutions). For S<or=6 he shows that these are the only solutions which are SU(2)-invariant, except for S=3 when there is a fifth solution.
In this thesis we theoretically investigate the close relation between spin chains and Yangians. First, we investigate the Haldane-Shastry model, which symmetry algebra is a representation of the Yangian. We show this in full detail for the spin-1/2 model. We also give an argument why a spin-1 chain with Yangian symmetry cannot exist. \nAfter that, we turn to integrable spin chains which are constructed using the quantum inverse scattering method. Using this method, an integrable spin chain can be constructed from each so called R-matrix, which is a solution of the Yang-Baxter equation. Our main result is the construction of an R-matrix that is invariant under the adjoint representation of SU(n), which is found using an intertwining operator for the corresponding Yangian representation. The integrable\nspin chain that corresponds to this R-matrix is non-Hermitian, making its physical interpretation unclear. These results have been published recently as a preprint under arXiv:1606.02516. \nFinally, integrable spin chains with defects are studied. We present the Hamiltonian for the Heisenberg model and the Babujian-Takhtajan model with defects, such that these models keep there integrability while staying Lie algebra invariant.
All regular 4×44 \times 44×4 solutions of the Yang-Baxter equation
In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times16 16 × 16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2) 𝔰 𝔲 ( 2 ) ⊕ 𝔰 𝔲 ( 2 ) symmetry, which include the one-dimensional Hubbard model and the S S -matrix of the {AdS}_5 \times {S}^5 A d S 5 × S 5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.